Twenty five coloured beads are to be arranged in a grid comprising of five rows and five columns. Each cell in the grid must contain exactly one bead. Each bead is coloured either Red, Blue or Green. While arranging the beads along any of the five rows or along any of the five columns, the rules given below are to be followed: (1) Two adjacent beads along the same row or column are always of different colours. (2) There is at least one Green bead between any two Blue beads along the same row or column. (3) There is at least one Blue and at least one Green bead between any two Red beads along the same row or column. Every unique, complete arrangement of twenty five beads is called a configuration.
Question: 1
The total number of possible configuration using beads of only two colours is:
As we need to use only two colours, in any row or column these two coloured beads will be placed alternately like
1
2
1
2
1
So we cannot place Red coloured beads at position 1 or two as between any two Red beads there must at least two beads (at least one green and at least one Blue). Hence, we can use only Green and Blue coloured beads.
We can have two possible configurations: Configuration 1: Green bead is placed at top left corner
G
B
G
B
G
B
G
B
G
B
G
B
G
B
G
B
G
B
G
B
G
B
G
B
G
Configuration 2: Blue bead is placed at top left corner
B
G
B
G
G
G
B
G
B
B
B
G
B
G
G
G
B
G
B
B
B
G
B
G
G
So, the answer is 2.
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Question: 2
What is the maximum possible number of Red beads that can appear in any configuration?
Between Any two Red beads there must be at least two Beads. So any Row or column there can be maximum two red beads. If we place two red beads in each row then two columns will have three red bead which cannot be accepted.
R
R
R
R
R
R
R
R
R
R
The above configuration is not correct. So, in the third row we will place only one Red bead at the middle of the third row. Also we will adjust other rows so that between any two Red beads there are at least two beads in any column.
R
R
R
R
R
R
R
R
R
So, maximum 9 Red beads are possible in any configuration. At remaining places Green and Blue coloured beads can be placed in such way that all the conditions given are satisfied. There are multiple configurations are possible. One of the configurations is given as below.
R
G
B
R
G
G
R
G
B
R
B
G
R
G
B
R
B
G
R
G
G
R
B
G
R
So, the answer is 9.
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Question: 3
What is the minimum number of Blue beads in any configuration?
To minimise number of Blue beads we need to maximise number of Red and Green beads. From the previous question solution, Maximum no. Red beads can be 9. The row in which has two red beads, we will place two green and one Blue bead additionally. The row with only one red bead we will place two green and two blue beads additionally. So overall there will be minimum 6 Blue beads.
R
G
B
R
G
G
R
G
B
R
B
G
R
G
B
R
B
G
R
G
G
R
B
G
R
So, the answer is 6.
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Question: 4
Two Red beads have been placed in ‘second row, third column’ and ‘third row, second column’. How many more Red beads can be placed so as to maximise the number of Red beads used in the configuration?
